Related papers: Additive and multiplicative structure of c$^{\star…

In this work we prove that the set of points at infinity $S_\infty:={\rm Cl}_{{\mathbb R}{\mathbb P}^m}(S)\cap\mathsf{H}_\infty$ of a semialgebraic set $S\subset{\mathbb R}^m$ which is the image of a polynomial map $f:{\mathbb…

We say the sets of nonnegative integers A and B are additive complements if their sum contains all sufficiently large integers. In this paper we prove a conjecture of Chen and Fang about additive complement of a finite set.

We review recent high spatial resolution millimeter continuum and spectral line observations of (proto-)cluster regions. These observations reveal that the mass distribution of prestellar cores is consistent with the initial mass function…

Let $0\le \alpha \le \beta\le 1$. For any finite set $B\subset\mathbb{N}$, we show that there exists a set $A\subset\mathbb{N}$ such that $\underline{d}(A+B) = \alpha$ and $\bar{d}(A+B) = \beta$, where $\underline{d}(A+ B)$ and…

Let $A$ be a subset of a finite field $\mathbb{F}$. When $\mathbb{F}$ has prime order, we show that there is an absolute constant $c > 0$ such that, if $A$ is both sum-free and equal to the set of its multiplicative inverses, then $|A| <…

We present a system of axioms motivated by a topological intuition: The set of subsets of any set is a topology on that set. On the one hand, this system is a common weakening of Zermelo-Fraenkel set theory ZF, the positive set theory GPK…

The nearest young stellar groups are associated with "hubs" of column density exceeding 10^22 cm^-2, according to recent observations. These hubs radiate multiple "filaments" of parsec length, having lower column density and fewer stars.…

We announce two breakthrough results concerning important questions in the Theory of Computational Complexity. In this expository paper, a systematic and comprehensive geometric characterization of the Subset Sum Problem is presented. We…

In this paper we continue our research of functions on the boundary of their domain and obtain some results on cluster sets of functions between topological spaces. In particular, we prove that for a metrizable topological space $X$, a…

In "IP-sets and polynomial recurrence", Bergelson, Furstenberg, and McCutcheon established the following far reaching extension of Khintchine's recurrence theorem: For any invertible probability preserving system $(X,\mathcal A,\mu,T)$, any…

Generalized topological spaces in the sense of Cs\'{a}sz\'{a}r have two main features which distinguish them from typical topologies. First, these families of subsets are not closed under intersections. Second, we allow for the possibility…

We construct an injective map from the set of holomorphic equivalence classes of neighborhoods $M$ of a compact complex manifold $C$ into ${\mathbb C}^m$ for some $m<\infty$ when $(TM)|_C$ is fixed and the normal bundle of $C$ in $M$ is…

Denote by $\mathbb{N}$ and $\mathbb{P}$ the set of all positive integers and prime numbers, respectively. Let $\mathbb{P}=\{p_1<p_2<\dots <p_n<\dots\}$, where $p_n$ is the $n$-th prime number. For $k\in\mathbb{N}$ we recursively define…

For a fixed $c > 0$ we construct an arbitrarily large set $B$ of size $n$ such that its sum set $B+B$ contains a convex sequence of size $cn^2$, answering a question of Hegarty.

A subset $A$ of $\nats$ is called an IP-set if $A$ contains all finite sums of distinct terms of some infinite sequence $(x_n)_{n\in \nats} $ of natural numbers. Central sets, first introduced by Furstenberg using notions from topological…

When defining the amount of additive structure on a set it is often convenient to consider certain sumsets; Calculating the cardinality of these sumsets can elucidate the set's underlying structure. We begin by investigating finite sets of…

In this paper, we are interested in the limit theorem question for sums of indicator functions. We show that in every aperiodic dynamical system, for every increasing sequence $(a_n)_{n\in\N}\subset\R_+$ such that $a_n\nearrow\infty$ and…

We discuss the mechanism of cluster formation in hierarchically collapsing molecular clouds. Recent evidence, both observational and numerical, suggests that molecular clouds (MCs) may be undergoing global, hierarchical gravitational…

We construct a spectral sequence associated to a stratified space, which computes the compactly supported cohomology groups of an open stratum in terms of the compactly supported cohomology groups of closed strata and the reduced cohomology…

Given a non-negative real sequence $\{c_n\}_n$ such that the series $\sum_{n=1}^{\infty}c_n$ diverges, it is known that the size of an infinite subset $A\subset\mathbb{N}$ can be measured in terms of the linear density such that the…